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1

The only criterion I know is based on a Theorem in the second book of Dunford and Schwartz, see Theorem X1.6.29 and following. If the resolvent of an Hilbert-Schimdt operator satisfies some decay estimates on some rays dividing the complex plane, then the span of the generalized eigenfuntions is dense. The theorem generalizes to operators having trace-class ...


0

In a finite-dimensional Hilbert space, this is the multivariate generalization of Stein's lemma, see Stein (1981), or lemma 1 in Liu (2003). You would need that $\partial f/\partial X_i$ exists almost everywhere and has finite expectation value. The case of an infinite-dimensional Hilbert space is treated in section 3 of Integration by parts formula and the ...


6

Injective envelopes are brutal. Let $A=\text{UHF}(2^\infty)$ and $B$ the hyperfinite II$_1$ factor. Take $f$ to be the inclusion map. We have $I(B)=B$, while $I(A)$ is a wild AW$^*$ factor of type III. If $g:I(A)\to B$ is a $*$-homomorphism and $\tau$ is the trace on $B$, then $\gamma=\tau\circ g$ is a trace on $I(A)$. In a type III AW$^*$-factor any ...


7

One can view $A$ and $B$ as sitting completely isometrically inside their injective envelopes $I(A)$ and $I(B)$. Then by injectivity a unital *-homomorphism (or more generally a unital completely positive map) $f:A\rightarrow B\subseteq I(B)$ extends to a unital completely positive map $\overline f:I(A) \rightarrow I(B)$. [Edit: this should work] Paulsen in ...


0

This is not a complete answer, but too long for a comment. Suppose that you have such a mapping $F$ between the unbounded s.a. operators. First consider its effect on multiples of the identity operator, say $\lambda I$. If it is implemented by such a real function $f$, then the image must be of the form $f(\lambda) I$. Hence, a first necessary condition ...


2

Miscellaneous results. If $A$ is strictly upper triangular, then $x\cdot\nabla$ consists only is terms $x_j\partial_k$ with $j<k$. The action of $L$ over homogenous polynomials of degree $d$ is described, in the basis of monomials written in lexicographic order, by a strictly upper triangular matrix. hence the only eigenvalue is $\lambda=0$. In general \...


0

A simple proof: Assume $\mathcal{H}=\mathcal{H}_+\oplus\mathcal{H}_-$, with $( A u| u)>0$ (resp. $<0$) for all nonzero $u\in\mathcal{H}_+$ (resp. $u\in\mathcal{H}_-$). We want to show $\ker A=0$. To this end, assume $Au=0$ and write $u=u_++u_-$ with $u_\pm\in\mathcal{H}_\pm$. If either $u_+$ or $u_-$ is zero, the assumption easily implies $u=0$. ...


3

Hamana's proof (Theorem 3.5 in Injective Envelopes of Operator Systems, PubL RIMS, Kyoto Univ. 15 (1979), 773-785) is fairly direct. Consider the partial ordering on the space $\Xi = \{ \phi \in UCP(W, W) \mid \phi \kappa = \kappa \}$ given by $\phi \prec \psi$ when $\| \phi(x) \| \leq \| \psi(x) \|$ for all $x \in W$. First, note that every element in $\Xi$ ...


1

Let $\varphi: W \rightarrow W$ be a u.c.p map. Let Fix$(\varphi)=\{ w \in W: \varphi(w)=w\}$. Then, Fix$(\varphi)$ is an injective operator system, if $W$ is an injective operator system. Moreover, Fix$(\varphi)$ contains $\kappa(V)$, as $\varphi \circ \kappa= \kappa$. By minimality of $W$, we must therefore have that Fix$(\varphi)= W$, thereby establishing ...


2

(TL, DR). Modulo the duality theory of Banach spaces, this simply amounts to null-sequences together with their limit, $0$, are compact, which is really trivial. But I agree that Takesaki probably could've made the exposition more clear by mentioning the Krein-Smulian theorem which is used implicitely so many times here. Below is the elaboration of the above....


9

Injectivity of $T$ plays no role. The point is that unital completely positive maps are contractive, so if a strict inequality $\|\phi(a)\|< \|a\|$ holds for some $a\in T$, then $\|a\|=\|\psi(\phi(a))\|\leqslant \|\phi(a)\| < \|a\|$, which is a contradiction. This shows that $\phi$ is an isometry and the same argument applied to matrix amplifications ...


6

I'm not sure if the following is exactly what the OP was looking for, but it definitely solves the question. The following lemma implies that $\mathcal O_2$ is a quotient of $C^*(\underbrace{(\mathbb Z/2\mathbb Z) * \dots * (\mathbb Z/2\mathbb Z)}_{6\textrm{ times}} )$. Lemma: Let $A$ be a unital $C^\ast$-algebra which is generated as a unital $C^\ast$-...


5

If one reads Takesaki's proof carefully, we find: and there (to my reading) is no claim that part (ii) uses "general duality theory". So let's look at Lemmas 2.4 and 2.5. 2.4 says that $\sigma$-strong$^\ast$ continuous functionals are $\sigma$-weakly continuous. 2.5 says (in particular) that on $S$ the $\sigma$-strong$^\ast$ and strong$^\ast$ ...


2

This is covered in the the monograph „Saks Spaces and Applications to Functional Analysis“. The short version is that one can use the fact that all closed subspaces (in particular, f.d. ones) of a Hilbert space are nicely complemented to express $L(H)$ as a projective limit of special cases of operators on finite dimensional spaces. There are three ways ...


0

I don’t think there’s any completely general method for doing so. In the case of the generalised Dirac operator $D$ induced by a Clifford connection $\nabla$ on a Clifford module bundle $E$ on a complete Riemannian manifold $X$ (e.g., the spin Dirac operator on a complete Riemannian spin manifold), you have a Lichnerowicz formula $$D^2 = \nabla^\ast \nabla + ...


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