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 $?