## oa.operator algebras – Lower bounds in the space of compact operators

Let $$H$$ be a separable Hilbert space, and $$K(H)$$ the corresponding space of compact operators. Consider the “unit sphere” $$S:={Tin K(H)|Tgeq 0text{ and }||T||=1}$$. Is it true that, given any pair of operators $$T_1,T_2in S$$, there exists another operator $$Tin S$$ such that $$Tleq T_1,T_2$$?.

## oa.operator algebras – Need reference for ideals and representations of \$C_0(X,A)\$

Let $$A$$ be $$C^{ast}$$– Algebra and $$X$$ be a locally compact Hausdorff space and $$C_{0}(X,A)$$ be the set of all continuous functions from $$X$$ to $$A$$ vanishing at infinity. Define $$f^{ast}(t)={f(t)}^{ast}$$ (for $$tin X$$). It is well known that $$C_0(X,A)$$ is $$C^{ast}-$$ Algebra.

What’s known about ideals and representations of $$C_0(X,A)$$?

My guess is that it must be related with ideals and representations of $$A$$. Can someone give a reference or some ideas?

P.S: The same question was first posted on MSE but unfortunately I dint not get any answer so I am posting it here.

## oa.operator algebras – Operator Space Tensioner Products

Given two Banach algebras $$A$$ Y $$B$$ with operator space structure in each of them, that is, both are closed subspaces of $$B (H_1)$$ Y $$B (H_2)$$ respectively for some Hilbert spaces $$H_1, H_2$$. Does your operator space the projective tensor product and Haagerup tensor products form Banach algebra? In other words, the operator space is projective / the Haagerup tensor standard is submultiplicative in $$A otimes B$$?