5a. Normed algebras

[math] \newcommand{\mathds}{\mathbb}[/math]

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We have seen that the study of the bounded operators [math]T\in B(H)[/math] often leads to the consideration of the algebras [math] \lt T \gt \subset B(H)[/math] generated by such operators, the idea being that the study of [math]A= \lt T \gt [/math] can lead to results about [math]T[/math] itself. In the remainder of this book we focus on the study of such algebras [math]A\subset B(H)[/math]. Before anything, we should mention that there are countless ways of getting introduced to operator algebras, depending on motivations and taste, with the available books including:

(1) The old book of von Neumann , which started everything. This is a very classical book, with mathematical physics content, written at times when mathematics and physics were starting to part ways. A great book, still enjoyable nowadays.

(2) Various post-war treatises, such as Dixmier , Kadison-Ringrose , Str\u atil\u a-Zsidò and Takesaki . As a warning, however, these books are purely mathematical. Also, they sometimes avoid deep results of von Neumann and Connes.

(3) More recent books, including Arveson , Blackadar , Brown-Ozawa , Connes , Davidson , Jones , Murphy , Pedersen and Sakai . These are well-concieved one-volume books, written with various purposes in mind.

Our presentation below is inspired by Blackadar , Connes , Jones , but is yet another type of beast, often insisting on probabilistic aspects. But probably enough talking, more on this later, and let us get to work. We are interested in the study of the algebras of bounded operators [math]A\subset B(H)[/math]. Let us start our discussion with the following broad definition, obtained by imposing the “minimal” set of reasonable axioms:

Definition

An operator algebra is an algebra of bounded operators [math]A\subset B(H)[/math] which contains the unit, is closed under taking adjoints,

[[math]] T\in A\implies T^*\in A [[/math]]

and is closed as well under the norm.

Here, as in the previous chapters, [math]H[/math] is an arbitrary Hilbert space, with the case that we are mostly interested in being the separable one. By separable we mean having a countable orthonormal basis, [math]\{e_i\}_{i\in I}[/math] with [math]I[/math] countable, and such a space is of course unique. The simplest model is the space [math]l^2(\mathbb N)[/math], but in practice, we are particularly interested in the spaces of the form [math]H=L^2(X)[/math], which are separable too, but with the basis [math]\{e_i\}_{i\in\mathbb N}[/math] and the subsequent identification [math]H\simeq l^2(\mathbb N)[/math] being not necessarily very explicit.

Also as in the previous chapters, [math]B(H)[/math] is the algebra of linear operators [math]T:H\to H[/math] which are bounded, in the sense that the norm [math]||T||=\sup_{||x||=1}||Tx||[/math] is finite. This algebra has an involution [math]T\to T^*[/math], with the adjoint operator [math]T^*\in B(H)[/math] being defined by the formula [math] \lt Tx,y \gt = \lt x,T^*y \gt [/math], and in the above definition, the assumption [math]T\in A\implies T^*\in A[/math] refers to this involution. Thus, [math]A[/math] must be a [math]*[/math]-algebra.

As a first result now regarding the operator algebras, in relation with the normal operators, where most of the non-trivial results that we have so far are, we have:

Theorem

The operator algebra [math] \lt T \gt \subset B(H)[/math] generated by a normal operator [math]T\in B(H)[/math] appears as an algebra of continuous functions,

[[math]] \lt T \gt =C(\sigma(T)) [[/math]]

where [math]\sigma(T)\subset\mathbb C[/math] denotes as usual the spectrum of [math]T[/math].

Show Proof

This is an abstract reformulation of the continuous functional calculus theorem for the normal operators, that we know from chapter 3. Indeed, that theorem tells us that we have a continuous morphism of [math]*[/math]-algebras, as follows:

[[math]] C(\sigma(T))\to B(H)\quad,\quad f\to f(T) [[/math]]

Moreover, by the general properties of the continuous calculus, also established in chapter 3, this morphism is injective, and its image is the norm closed algebra [math] \lt T \gt [/math] generated by [math]T,T^*[/math]. Thus, we obtain the isomorphism in the statement.

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The above result is very nice, and it is possible to further build on it, by using this time the spectral theorem for families of normal operators, as follows:

Theorem

The operator algebra [math] \lt T_i \gt \subset B(H)[/math] generated by a family of normal operators [math]T_i\in B(H)[/math] appears as an algebra of continuous functions,

[[math]] \lt T \gt =C(X) [[/math]]

where [math]X\subset\mathbb C[/math] is a certain compact space associated to the family [math]\{T_i\}[/math]. Equivalently, any commutative operator algebra [math]A\subset B(H)[/math] is of the form [math]A=C(X)[/math].

Show Proof

We have two assertions here, the idea being as follows:

(1) Regarding the first assertion, this follows exactly as in the proof of Theorem 5.2, by using this time the spectral theorem for families of normal operators.

(2) As for the second assertion, this is clear from the first one, because any commutative algebra [math]A\subset B(H)[/math] is generated by its elements [math]T\in A[/math], which are all normal.

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All this is good to know, but Theorem 5.2 and Theorem 5.3 remain something quite heavy, based on the spectral theorem. We would like to present now an alternative proof for these results, which is rather elementary, and has the advantage of reconstructing the compact space [math]X[/math] directly from the knowledge of the algebra [math]A[/math]. We will need:

Theorem

Given an operator [math]T\in A\subset B(H)[/math], define its spectrum as:

[[math]] \sigma(T)=\left\{\lambda\in\mathbb C\Big|T-\lambda\notin A^{-1}\right\} [[/math]]

The following spectral theory results hold, exactly as in the [math]A=B(H)[/math] case:

• We have [math]\sigma(ST)\cup\{0\}=\sigma(TS)\cup\{0\}[/math].
• We have polynomial, rational and holomorphic calculus.
• As a consequence, the spectra are compact and non-empty.
• The spectra of unitaries [math](U^*=U^{-1})[/math] and self-adjoints [math](T=T^*)[/math] are on [math]\mathbb T,\mathbb R[/math].
• The spectral radius of normal elements [math](TT^*=T^*T)[/math] is given by [math]\rho(T)=||T||[/math].

In addition, assuming [math]T\in A\subset B[/math], the spectra of [math]T[/math] with respect to [math]A[/math] and to [math]B[/math] coincide.

Show Proof

This is something that we know from the beginning of chapter 3, in the case [math]A=B(H)[/math]. In general the proof is similar, the idea being as follows:

(1) Regarding the assertions (1-5), which are of course formulated a bit informally, the proofs here are perfectly similar to those for the full operator algebra [math]A=B(H)[/math]. All this is standard material, and in fact, things in chapter 3 were written in such a way as for their extension now, to the general operator algebra setting, to be obvious.

(2) Regarding the last assertion, the inclusion [math]\sigma_B(T)\subset\sigma_A(T)[/math] is clear. For the converse, assume [math]T-\lambda\in B^{-1}[/math], and consider the following self-adjoint element:

[[math]] S=(T-\lambda )^*(T-\lambda ) [[/math]]

The difference between the two spectra of [math]S\in A\subset B[/math] is then given by:

[[math]] \sigma_A(S)-\sigma_B(S)=\left\{\mu\in\mathbb C-\sigma_B(S)\Big|(S-\mu)^{-1}\in B-A\right\} [[/math]]

Thus this difference in an open subset of [math]\mathbb C[/math]. On the other hand [math]S[/math] being self-adjoint, its two spectra are both real, and so is their difference. Thus the two spectra of [math]S[/math] are equal, and in particular [math]S[/math] is invertible in [math]A[/math], and so [math]T-\lambda\in A^{-1}[/math], as desired.

(3) As an observation, the last assertion applied with [math]B=B(H)[/math] shows that the spectrum [math]\sigma(T)[/math] as constructed in the statement coincides with the spectrum [math]\sigma(T)[/math] as constructed and studied in chapter 3, so the fact that (1-5) hold indeed is no surprise.

(4) Finally, I can hear you screaming that I should have concieved this book differently, matter of not proving the same things twice. Good point, with my distinguished colleague Bourbaki saying the same, and in answer, wait for chapter 7 below, where we will prove exactly the same things a third time. We can discuss pedagogy at that time.

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We can now get back to the commutative algebras, and we have the following result, due to Gelfand, which provides an alternative to Theorem 5.2 and Theorem 5.3:

Theorem

Any commutative operator algebra [math]A\subset B(H)[/math] is of the form

[[math]] A=C(X) [[/math]]

with the “spectrum” [math]X[/math] of such an algebra being the space of characters [math]\chi:A\to\mathbb C[/math], with topology making continuous the evaluation maps [math]ev_T:\chi\to\chi(T)[/math].

Show Proof

Given a commutative operator algebra [math]A[/math], we can define [math]X[/math] as in the statement. Then [math]X[/math] is compact, and [math]T\to ev_T[/math] is a morphism of algebras, as follows:

[[math]] ev:A\to C(X) [[/math]]

(1) We first prove that [math]ev[/math] is involutive. We use the following formula, which is similar to the [math]z=Re(z)+iIm(z)[/math] formula for the usual complex numbers:

[[math]] T=\frac{T+T^*}{2}+i\cdot\frac{T-T^*}{2i} [[/math]]

Thus it is enough to prove the equality [math]ev_{T^*}=ev_T^*[/math] for self-adjoint elements [math]T[/math]. But this is the same as proving that [math]T=T^*[/math] implies that [math]ev_T[/math] is a real function, which is in turn true, because [math]ev_T(\chi)=\chi(T)[/math] is an element of [math]\sigma(T)[/math], contained in [math]\mathbb R[/math].

(2) Since [math]A[/math] is commutative, each element is normal, so [math]ev[/math] is isometric:

[[math]] ||ev_T|| =\rho(T) =||T|| [[/math]]

(3) It remains to prove that [math]ev[/math] is surjective. But this follows from the Stone-Weierstrass theorem, because [math]ev(A)[/math] is a closed subalgebra of [math]C(X)[/math], which separates the points.

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The above theorem of Gelfand is something very beautiful, and far-reaching. It is possible to further build on it, indefinitely high. We will be back to this.

General references

Banica, Teo (2024). "Principles of operator algebras". arXiv:2208.03600 [math.OA].